Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  gsumhashmul Structured version   Visualization version   GIF version

Theorem gsumhashmul 30744
 Description: Express a group sum by grouping by nonzero values. (Contributed by Thierry Arnoux, 22-Jun-2024.)
Hypotheses
Ref Expression
gsumhashmul.b 𝐵 = (Base‘𝐺)
gsumhashmul.z 0 = (0g𝐺)
gsumhashmul.x · = (.g𝐺)
gsumhashmul.g (𝜑𝐺 ∈ CMnd)
gsumhashmul.f (𝜑𝐹:𝐴𝐵)
gsumhashmul.1 (𝜑𝐹 finSupp 0 )
Assertion
Ref Expression
gsumhashmul (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥))))
Distinct variable groups:   𝑥, 0   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥
Allowed substitution hint:   · (𝑥)

Proof of Theorem gsumhashmul
Dummy variables 𝑡 𝑢 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumhashmul.f . . . . . . 7 (𝜑𝐹:𝐴𝐵)
2 suppssdm 7830 . . . . . . . 8 (𝐹 supp 0 ) ⊆ dom 𝐹
32, 1fssdm 6508 . . . . . . 7 (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
41, 3feqresmpt 6713 . . . . . 6 (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑥)))
54oveq2d 7155 . . . . 5 (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝐺 Σg (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑥))))
6 gsumhashmul.b . . . . . 6 𝐵 = (Base‘𝐺)
7 gsumhashmul.z . . . . . 6 0 = (0g𝐺)
8 gsumhashmul.g . . . . . 6 (𝜑𝐺 ∈ CMnd)
9 gsumhashmul.1 . . . . . . . 8 (𝜑𝐹 finSupp 0 )
10 relfsupp 8823 . . . . . . . . 9 Rel finSupp
1110brrelex1i 5576 . . . . . . . 8 (𝐹 finSupp 0𝐹 ∈ V)
129, 11syl 17 . . . . . . 7 (𝜑𝐹 ∈ V)
131ffnd 6492 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
1412, 13fndmexd 7601 . . . . . 6 (𝜑𝐴 ∈ V)
15 ssidd 3941 . . . . . 6 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
166, 7, 8, 14, 1, 15, 9gsumres 19029 . . . . 5 (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝐺 Σg 𝐹))
17 nfcv 2958 . . . . . 6 𝑥(𝐹‘(1st𝑧))
18 fveq2 6649 . . . . . 6 (𝑥 = (1st𝑧) → (𝐹𝑥) = (𝐹‘(1st𝑧)))
199fsuppimpd 8828 . . . . . 6 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
20 ssidd 3941 . . . . . 6 (𝜑𝐵𝐵)
211adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝐹:𝐴𝐵)
223sselda 3918 . . . . . . 7 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝑥𝐴)
2321, 22ffvelrnd 6833 . . . . . 6 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → (𝐹𝑥) ∈ 𝐵)
241ffund 6495 . . . . . . . . 9 (𝜑 → Fun 𝐹)
25 funrel 6345 . . . . . . . . 9 (Fun 𝐹 → Rel 𝐹)
26 reldif 5656 . . . . . . . . 9 (Rel 𝐹 → Rel (𝐹 ∖ (V × { 0 })))
2724, 25, 263syl 18 . . . . . . . 8 (𝜑 → Rel (𝐹 ∖ (V × { 0 })))
28 1stdm 7725 . . . . . . . 8 ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st𝑧) ∈ dom (𝐹 ∖ (V × { 0 })))
2927, 28sylan 583 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st𝑧) ∈ dom (𝐹 ∖ (V × { 0 })))
307fvexi 6663 . . . . . . . . . . . 12 0 ∈ V
3130a1i 11 . . . . . . . . . . 11 (𝜑0 ∈ V)
32 fressupp 30451 . . . . . . . . . . 11 ((Fun 𝐹𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
3324, 12, 31, 32syl3anc 1368 . . . . . . . . . 10 (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
3433dmeqd 5742 . . . . . . . . 9 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = dom (𝐹 ∖ (V × { 0 })))
352a1i 11 . . . . . . . . . 10 (𝜑 → (𝐹 supp 0 ) ⊆ dom 𝐹)
36 ssdmres 5845 . . . . . . . . . 10 ((𝐹 supp 0 ) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
3735, 36sylib 221 . . . . . . . . 9 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
3834, 37eqtr3d 2838 . . . . . . . 8 (𝜑 → dom (𝐹 ∖ (V × { 0 })) = (𝐹 supp 0 ))
3938adantr 484 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → dom (𝐹 ∖ (V × { 0 })) = (𝐹 supp 0 ))
4029, 39eleqtrd 2895 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st𝑧) ∈ (𝐹 supp 0 ))
4124funresd 6371 . . . . . . . . . . 11 (𝜑 → Fun (𝐹 ↾ (𝐹 supp 0 )))
4241adantr 484 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → Fun (𝐹 ↾ (𝐹 supp 0 )))
4337eleq2d 2878 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑥 ∈ (𝐹 supp 0 )))
4443biimpar 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )))
45 simpr 488 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ (𝐹 supp 0 ))
4645fvresd 6669 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥))
47 funopfvb 6700 . . . . . . . . . . 11 ((Fun (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 ))))
4847biimpa 480 . . . . . . . . . 10 (((Fun (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥)) → ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 )))
4942, 44, 46, 48syl21anc 836 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 )))
5033adantr 484 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
5149, 50eleqtrd 2895 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ∖ (V × { 0 })))
52 eqeq2 2813 . . . . . . . . . . 11 (𝑣 = ⟨𝑥, (𝐹𝑥)⟩ → (𝑧 = 𝑣𝑧 = ⟨𝑥, (𝐹𝑥)⟩))
5352bibi2d 346 . . . . . . . . . 10 (𝑣 = ⟨𝑥, (𝐹𝑥)⟩ → ((𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣) ↔ (𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)))
5453ralbidv 3165 . . . . . . . . 9 (𝑣 = ⟨𝑥, (𝐹𝑥)⟩ → (∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣) ↔ ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)))
5554adantl 485 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑣 = ⟨𝑥, (𝐹𝑥)⟩) → (∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣) ↔ ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)))
56 fvexd 6664 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → (2nd𝑧) ∈ V)
5727ad3antrrr 729 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → Rel (𝐹 ∖ (V × { 0 })))
58 simplr 768 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
59 1st2nd 7724 . . . . . . . . . . . . . . . . 17 ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
6057, 58, 59syl2anc 587 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
61 opeq1 4766 . . . . . . . . . . . . . . . . 17 (𝑥 = (1st𝑧) → ⟨𝑥, (2nd𝑧)⟩ = ⟨(1st𝑧), (2nd𝑧)⟩)
6261adantl 485 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → ⟨𝑥, (2nd𝑧)⟩ = ⟨(1st𝑧), (2nd𝑧)⟩)
6360, 62eqtr4d 2839 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 = ⟨𝑥, (2nd𝑧)⟩)
64 difssd 4063 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → (𝐹 ∖ (V × { 0 })) ⊆ 𝐹)
6564sselda 3918 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧𝐹)
6665adantr 484 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧𝐹)
6763, 66eqeltrrd 2894 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹)
6863, 67jca 515 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → (𝑧 = ⟨𝑥, (2nd𝑧)⟩ ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹))
69 opeq2 4768 . . . . . . . . . . . . . . . 16 (𝑦 = (2nd𝑧) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, (2nd𝑧)⟩)
7069eqeq2d 2812 . . . . . . . . . . . . . . 15 (𝑦 = (2nd𝑧) → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, (2nd𝑧)⟩))
7169eleq1d 2877 . . . . . . . . . . . . . . 15 (𝑦 = (2nd𝑧) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹))
7270, 71anbi12d 633 . . . . . . . . . . . . . 14 (𝑦 = (2nd𝑧) → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (𝑧 = ⟨𝑥, (2nd𝑧)⟩ ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹)))
7356, 68, 72spcedv 3550 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
74 vex 3447 . . . . . . . . . . . . . 14 𝑥 ∈ V
7574elsnres 5862 . . . . . . . . . . . . 13 (𝑧 ∈ (𝐹 ↾ {𝑥}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
7673, 75sylibr 237 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 ∈ (𝐹 ↾ {𝑥}))
7713ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝐹 Fn 𝐴)
7822ad2antrr 725 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑥𝐴)
79 fnressn 6901 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
8077, 78, 79syl2anc 587 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
8176, 80eleqtrd 2895 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 ∈ {⟨𝑥, (𝐹𝑥)⟩})
82 elsni 4545 . . . . . . . . . . 11 (𝑧 ∈ {⟨𝑥, (𝐹𝑥)⟩} → 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)
8381, 82syl 17 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)
84 simpr 488 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩) → 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)
8584fveq2d 6653 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩) → (1st𝑧) = (1st ‘⟨𝑥, (𝐹𝑥)⟩))
86 fvex 6662 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
8774, 86op1st 7683 . . . . . . . . . . 11 (1st ‘⟨𝑥, (𝐹𝑥)⟩) = 𝑥
8885, 87eqtr2di 2853 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩) → 𝑥 = (1st𝑧))
8983, 88impbida 800 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩))
9089ralrimiva 3152 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩))
9151, 55, 90rspcedvd 3577 . . . . . . 7 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ∃𝑣 ∈ (𝐹 ∖ (V × { 0 }))∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣))
92 reu6 3668 . . . . . . 7 (∃!𝑧 ∈ (𝐹 ∖ (V × { 0 }))𝑥 = (1st𝑧) ↔ ∃𝑣 ∈ (𝐹 ∖ (V × { 0 }))∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣))
9391, 92sylibr 237 . . . . . 6 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ∃!𝑧 ∈ (𝐹 ∖ (V × { 0 }))𝑥 = (1st𝑧))
9417, 6, 7, 18, 8, 19, 20, 23, 40, 93gsummptf1o 19079 . . . . 5 (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑥))) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧)))))
955, 16, 943eqtr3d 2844 . . . 4 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧)))))
96 simpr 488 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
9796eldifad 3896 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧𝐹)
98 funfv1st2nd 7731 . . . . . . 7 ((Fun 𝐹𝑧𝐹) → (𝐹‘(1st𝑧)) = (2nd𝑧))
9924, 97, 98syl2an2r 684 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (𝐹‘(1st𝑧)) = (2nd𝑧))
10099mpteq2dva 5128 . . . . 5 (𝜑 → (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧))) = (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧)))
101100oveq2d 7155 . . . 4 (𝜑 → (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧)))) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧))))
10295, 101eqtrd 2836 . . 3 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧))))
103 nfcv 2958 . . . 4 𝑧(1st𝑡)
104 fvex 6662 . . . . 5 (2nd𝑡) ∈ V
105 fvex 6662 . . . . 5 (1st𝑡) ∈ V
106104, 105op2ndd 7686 . . . 4 (𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ → (2nd𝑧) = (1st𝑡))
107 resfnfinfin 8792 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝐹 supp 0 ) ∈ Fin) → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
10813, 19, 107syl2anc 587 . . . . 5 (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
10933, 108eqeltrrd 2894 . . . 4 (𝜑 → (𝐹 ∖ (V × { 0 })) ∈ Fin)
11033rneqd 5776 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = ran (𝐹 ∖ (V × { 0 })))
111 rnresss 5858 . . . . . 6 ran (𝐹 ↾ (𝐹 supp 0 )) ⊆ ran 𝐹
1121frnd 6498 . . . . . 6 (𝜑 → ran 𝐹𝐵)
113111, 112sstrid 3929 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) ⊆ 𝐵)
114110, 113eqsstrrd 3957 . . . 4 (𝜑 → ran (𝐹 ∖ (V × { 0 })) ⊆ 𝐵)
115 2ndrn 7726 . . . . 5 ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (2nd𝑧) ∈ ran (𝐹 ∖ (V × { 0 })))
11627, 115sylan 583 . . . 4 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (2nd𝑧) ∈ ran (𝐹 ∖ (V × { 0 })))
117 relcnv 5938 . . . . . . . 8 Rel 𝐹
118 reldif 5656 . . . . . . . 8 (Rel 𝐹 → Rel (𝐹 ∖ ({ 0 } × V)))
119117, 118mp1i 13 . . . . . . 7 (𝜑 → Rel (𝐹 ∖ ({ 0 } × V)))
120 1st2nd 7724 . . . . . . 7 ((Rel (𝐹 ∖ ({ 0 } × V)) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → 𝑡 = ⟨(1st𝑡), (2nd𝑡)⟩)
121119, 120sylan 583 . . . . . 6 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → 𝑡 = ⟨(1st𝑡), (2nd𝑡)⟩)
122 cnvdif 5973 . . . . . . . . . 10 (𝐹 ∖ (V × { 0 })) = (𝐹(V × { 0 }))
123 cnvxp 5985 . . . . . . . . . . 11 (V × { 0 }) = ({ 0 } × V)
124123difeq2i 4050 . . . . . . . . . 10 (𝐹(V × { 0 })) = (𝐹 ∖ ({ 0 } × V))
125122, 124eqtri 2824 . . . . . . . . 9 (𝐹 ∖ (V × { 0 })) = (𝐹 ∖ ({ 0 } × V))
126125eqimss2i 3977 . . . . . . . 8 (𝐹 ∖ ({ 0 } × V)) ⊆ (𝐹 ∖ (V × { 0 }))
127126a1i 11 . . . . . . 7 (𝜑 → (𝐹 ∖ ({ 0 } × V)) ⊆ (𝐹 ∖ (V × { 0 })))
128127sselda 3918 . . . . . 6 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → 𝑡(𝐹 ∖ (V × { 0 })))
129121, 128eqeltrrd 2894 . . . . 5 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
130105, 104opelcnv 5720 . . . . 5 (⟨(1st𝑡), (2nd𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })) ↔ ⟨(2nd𝑡), (1st𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
131129, 130sylib 221 . . . 4 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → ⟨(2nd𝑡), (1st𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
13227adantr 484 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → Rel (𝐹 ∖ (V × { 0 })))
133 eqidd 2802 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → {𝑧} = {𝑧})
134 cnvf1olem 7792 . . . . . . . . 9 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ {𝑧} = {𝑧})) → ( {𝑧} ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑧 = { {𝑧}}))
135134simpld 498 . . . . . . . 8 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ {𝑧} = {𝑧})) → {𝑧} ∈ (𝐹 ∖ (V × { 0 })))
136132, 96, 133, 135syl12anc 835 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → {𝑧} ∈ (𝐹 ∖ (V × { 0 })))
137136, 125eleqtrdi 2903 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → {𝑧} ∈ (𝐹 ∖ ({ 0 } × V)))
138 eqeq2 2813 . . . . . . . . 9 (𝑢 = {𝑧} → (𝑡 = 𝑢𝑡 = {𝑧}))
139138bibi2d 346 . . . . . . . 8 (𝑢 = {𝑧} → ((𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ (𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧})))
140139ralbidv 3165 . . . . . . 7 (𝑢 = {𝑧} → (∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ ∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧})))
141140adantl 485 . . . . . 6 (((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑢 = {𝑧}) → (∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ ∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧})))
142117, 118mp1i 13 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → Rel (𝐹 ∖ ({ 0 } × V)))
143 simplr 768 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑡 ∈ (𝐹 ∖ ({ 0 } × V)))
144 simpr 488 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩)
145 df-rel 5530 . . . . . . . . . . . . . 14 (Rel (𝐹 ∖ ({ 0 } × V)) ↔ (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
146119, 145sylib 221 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
147146ad3antrrr 729 . . . . . . . . . . . 12 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
148147, 143sseldd 3919 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑡 ∈ (V × V))
149 2nd1st 7723 . . . . . . . . . . 11 (𝑡 ∈ (V × V) → {𝑡} = ⟨(2nd𝑡), (1st𝑡)⟩)
150148, 149syl 17 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → {𝑡} = ⟨(2nd𝑡), (1st𝑡)⟩)
151144, 150eqtr4d 2839 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑧 = {𝑡})
152 cnvf1olem 7792 . . . . . . . . . 10 ((Rel (𝐹 ∖ ({ 0 } × V)) ∧ (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ∧ 𝑧 = {𝑡})) → (𝑧(𝐹 ∖ ({ 0 } × V)) ∧ 𝑡 = {𝑧}))
153152simprd 499 . . . . . . . . 9 ((Rel (𝐹 ∖ ({ 0 } × V)) ∧ (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ∧ 𝑧 = {𝑡})) → 𝑡 = {𝑧})
154142, 143, 151, 153syl12anc 835 . . . . . . . 8 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑡 = {𝑧})
15527ad3antrrr 729 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → Rel (𝐹 ∖ (V × { 0 })))
15696ad2antrr 725 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
157 simpr 488 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑡 = {𝑧})
158 cnvf1olem 7792 . . . . . . . . . . 11 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑡 = {𝑧})) → (𝑡(𝐹 ∖ (V × { 0 })) ∧ 𝑧 = {𝑡}))
159158simprd 499 . . . . . . . . . 10 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑡 = {𝑧})) → 𝑧 = {𝑡})
160155, 156, 157, 159syl12anc 835 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑧 = {𝑡})
161146ad3antrrr 729 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
162 simplr 768 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑡 ∈ (𝐹 ∖ ({ 0 } × V)))
163161, 162sseldd 3919 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑡 ∈ (V × V))
164163, 149syl 17 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → {𝑡} = ⟨(2nd𝑡), (1st𝑡)⟩)
165160, 164eqtrd 2836 . . . . . . . 8 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩)
166154, 165impbida 800 . . . . . . 7 (((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → (𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧}))
167166ralrimiva 3152 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧}))
168137, 141, 167rspcedvd 3577 . . . . 5 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∃𝑢 ∈ (𝐹 ∖ ({ 0 } × V))∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢))
169 reu6 3668 . . . . 5 (∃!𝑡 ∈ (𝐹 ∖ ({ 0 } × V))𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ ∃𝑢 ∈ (𝐹 ∖ ({ 0 } × V))∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢))
170168, 169sylibr 237 . . . 4 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∃!𝑡 ∈ (𝐹 ∖ ({ 0 } × V))𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩)
171103, 6, 7, 106, 8, 109, 114, 116, 131, 170gsummptf1o 19079 . . 3 (𝜑 → (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧))) = (𝐺 Σg (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡))))
172 fveq2 6649 . . . . . 6 (𝑡 = 𝑧 → (1st𝑡) = (1st𝑧))
173172cbvmptv 5136 . . . . 5 (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡)) = (𝑧 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑧))
17433cnveqd 5714 . . . . . . 7 (𝜑(𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
175174, 125eqtr2di 2853 . . . . . 6 (𝜑 → (𝐹 ∖ ({ 0 } × V)) = (𝐹 ↾ (𝐹 supp 0 )))
176175mpteq1d 5122 . . . . 5 (𝜑 → (𝑧 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑧)) = (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧)))
177173, 176syl5eq 2848 . . . 4 (𝜑 → (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡)) = (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧)))
178177oveq2d 7155 . . 3 (𝜑 → (𝐺 Σg (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡))) = (𝐺 Σg (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧))))
179102, 171, 1783eqtrd 2840 . 2 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧))))
180 nfcv 2958 . . 3 𝑦(1st𝑧)
181 nfv 1915 . . 3 𝑥𝜑
182 vex 3447 . . . 4 𝑦 ∈ V
18374, 182op1std 7685 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
184 relcnv 5938 . . . 4 Rel (𝐹 ↾ (𝐹 supp 0 ))
185184a1i 11 . . 3 (𝜑 → Rel (𝐹 ↾ (𝐹 supp 0 )))
186 cnvfi 8794 . . . 4 ((𝐹 ↾ (𝐹 supp 0 )) ∈ Fin → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
187108, 186syl 17 . . 3 (𝜑(𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
188112adantr 484 . . . 4 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → ran 𝐹𝐵)
189184a1i 11 . . . . . . 7 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → Rel (𝐹 ↾ (𝐹 supp 0 )))
190 simpr 488 . . . . . . 7 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → 𝑧(𝐹 ↾ (𝐹 supp 0 )))
191 1stdm 7725 . . . . . . 7 ((Rel (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ dom (𝐹 ↾ (𝐹 supp 0 )))
192189, 190, 191syl2anc 587 . . . . . 6 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ dom (𝐹 ↾ (𝐹 supp 0 )))
193 df-rn 5534 . . . . . 6 ran (𝐹 ↾ (𝐹 supp 0 )) = dom (𝐹 ↾ (𝐹 supp 0 ))
194192, 193eleqtrrdi 2904 . . . . 5 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ ran (𝐹 ↾ (𝐹 supp 0 )))
195111, 194sseldi 3916 . . . 4 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ ran 𝐹)
196188, 195sseldd 3919 . . 3 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ 𝐵)
197180, 181, 6, 183, 185, 187, 8, 196gsummpt2d 30737 . 2 (𝜑 → (𝐺 Σg (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧))) = (𝐺 Σg (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)))))
198 df-ima 5536 . . . . . . 7 (𝐹 “ (𝐹 supp 0 )) = ran (𝐹 ↾ (𝐹 supp 0 ))
199 supppreima 30454 . . . . . . . . 9 ((Fun 𝐹𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (ran 𝐹 ∖ { 0 })))
20024, 12, 31, 199syl3anc 1368 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (ran 𝐹 ∖ { 0 })))
201200imaeq2d 5900 . . . . . . 7 (𝜑 → (𝐹 “ (𝐹 supp 0 )) = (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))))
202198, 201syl5eqr 2850 . . . . . 6 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))))
203 funimacnv 6409 . . . . . . 7 (Fun 𝐹 → (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))) = ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹))
20424, 203syl 17 . . . . . 6 (𝜑 → (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))) = ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹))
205 difssd 4063 . . . . . . 7 (𝜑 → (ran 𝐹 ∖ { 0 }) ⊆ ran 𝐹)
206 df-ss 3901 . . . . . . 7 ((ran 𝐹 ∖ { 0 }) ⊆ ran 𝐹 ↔ ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹) = (ran 𝐹 ∖ { 0 }))
207205, 206sylib 221 . . . . . 6 (𝜑 → ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹) = (ran 𝐹 ∖ { 0 }))
208202, 204, 2073eqtrd 2840 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
209193, 208syl5eqr 2850 . . . 4 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
2108cmnmndd 18924 . . . . . . 7 (𝜑𝐺 ∈ Mnd)
211210adantr 484 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
212108adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
213 imafi2 30476 . . . . . . 7 ((𝐹 ↾ (𝐹 supp 0 )) ∈ Fin → ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin)
214212, 186, 2133syl 18 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin)
215193, 113eqsstrrid 3967 . . . . . . 7 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) ⊆ 𝐵)
216215sselda 3918 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → 𝑥𝐵)
217 gsumhashmul.x . . . . . . 7 · = (.g𝐺)
2186, 217gsumconst 19050 . . . . . 6 ((𝐺 ∈ Mnd ∧ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin ∧ 𝑥𝐵) → (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
219211, 214, 216, 218syl3anc 1368 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
220 cnvresima 6058 . . . . . . . 8 ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) = ((𝐹 “ {𝑥}) ∩ (𝐹 supp 0 ))
221209eleq2d 2878 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑥 ∈ (ran 𝐹 ∖ { 0 })))
222221biimpa 480 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → 𝑥 ∈ (ran 𝐹 ∖ { 0 }))
223222snssd 4705 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → {𝑥} ⊆ (ran 𝐹 ∖ { 0 }))
224 sspreima 30409 . . . . . . . . . . 11 ((Fun 𝐹 ∧ {𝑥} ⊆ (ran 𝐹 ∖ { 0 })) → (𝐹 “ {𝑥}) ⊆ (𝐹 “ (ran 𝐹 ∖ { 0 })))
22524, 223, 224syl2an2r 684 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 “ {𝑥}) ⊆ (𝐹 “ (ran 𝐹 ∖ { 0 })))
226200adantr 484 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 supp 0 ) = (𝐹 “ (ran 𝐹 ∖ { 0 })))
227225, 226sseqtrrd 3959 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 “ {𝑥}) ⊆ (𝐹 supp 0 ))
228 df-ss 3901 . . . . . . . . 9 ((𝐹 “ {𝑥}) ⊆ (𝐹 supp 0 ) ↔ ((𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) = (𝐹 “ {𝑥}))
229227, 228sylib 221 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → ((𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) = (𝐹 “ {𝑥}))
230220, 229syl5req 2849 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 “ {𝑥}) = ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}))
231230fveq2d 6653 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (♯‘(𝐹 “ {𝑥})) = (♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})))
232231oveq1d 7154 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → ((♯‘(𝐹 “ {𝑥})) · 𝑥) = ((♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
233219, 232eqtr4d 2839 . . . 4 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘(𝐹 “ {𝑥})) · 𝑥))
234209, 233mpteq12dva 5117 . . 3 (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥))) = (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥)))
235234oveq2d 7155 . 2 (𝜑 → (𝐺 Σg (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)))) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥))))
236179, 197, 2353eqtrd 2840 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2112  ∀wral 3109  ∃wrex 3110  ∃!wreu 3111  Vcvv 3444   ∖ cdif 3881   ∩ cin 3883   ⊆ wss 3884  {csn 4528  ⟨cop 4534  ∪ cuni 4803   class class class wbr 5033   ↦ cmpt 5113   × cxp 5521  ◡ccnv 5522  dom cdm 5523  ran crn 5524   ↾ cres 5525   “ cima 5526  Rel wrel 5528  Fun wfun 6322   Fn wfn 6323  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674   supp csupp 7817  Fincfn 8496   finSupp cfsupp 8821  ♯chash 13690  Basecbs 16478  0gc0g 16708   Σg cgsu 16709  Mndcmnd 17906  .gcmg 18219  CMndccmn 18901 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033  df-seq 13369  df-hash 13691  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-0g 16710  df-gsum 16711  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-submnd 17952  df-mulg 18220  df-cntz 18442  df-cmn 18903 This theorem is referenced by:  elrspunidl  31017
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